Terminal boundary-value technique for solving singularly perturbed delay differential equations

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Terminal boundary-value technique for solving singularly perturbed delay differential equations Gemechis File ∗,1 , Yanala Narsimha Reddy Department of Mathematics, National Institute of Technology, Warangal 506 004, India

Abstract A terminal boundary-value technique is presented for solving singularly perturbed delay differential equations, the solutions of which exhibit layer behaviour. By introducing a terminal point, the original problem is divided into inner and outer region problems. An implicit terminal boundary condition at the terminal point was determined. The outer region problem with the implicit boundary condition was solved and produces an explicit boundary condition for the inner region problem. Then, the modified inner region problem (using the stretching transformation) is solved as a two-point boundary value problem. The second-order finite difference scheme was used to solve both the inner and outer region problems. The proposed method is iterative on the terminal point. To validate the efficiency of the method, some model examples were solved. The stability and convergence of the scheme was also investigated. © 2014 Taibah University. Production and hosting by Elsevier B.V. All rights reserved. MSC: 65L03; 65L10; 65L11; 65L12 Keywords: Singular perturbation; Delay differential equations; Finite differences; Terminal boundary condition; Boundary layer

1. Introduction Singular perturbation problems containing a small parameter, ε, multiplying to their highest derivative term arise in many fields, such as fluid mechanics, fluid dynamics, chemical reactor theory and elasticity, which have received significant attention. The solution of these types of problems shows a multi-scale character, with a ∗

Corresponding author. Tel.: +91 9700325821. E-mail addresses: [email protected], [email protected] (G. File). 1 Permanent address: Jimma University, PO Box 378, Jimma, Ethiopia. Peer review under responsibility of Taibah University

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narrow region called the boundary layer, in which their solution changes rapidly, and an outer region in which the solution changes smoothly. Thus, the treatment of such problems is not trivial because of the boundary layer behaviour of their solutions. Detailed theory and analytical discussions of solving singular perturbation problems have been published [1–9], and have the details of numerical and asymptotic solutions [10–15]. Boundary value problems involving delay differential equations arise in studying the mathematical modelling of various practical phenomena, like micro-scale heat transfer [16], the hydrodynamics of liquid helium [17], second-sound theory [18], optically bi-stable devices [19], diffusion in polymers [20], reaction–diffusion equations [21], stability [22], control including control of chaotic systems [23] and a variety of models for physiological processes and diseases [24,25]. For example, Lange and Miura [26–30] treated the singular perturbation analysis of the problem under consideration in a series of papers. A numerical method based on the fitted mesh approach to approximate the solution

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of these types of boundary value problems was published by Kadalbajoo and Sharma [31]. The authors constructed piecewise-uniform meshes and fitted them to the boundary layer regions to adapt the singular behaviour of the operator in the narrow regions. The same authors [32] approximated the terms with delay by first-order Taylor series expansions to analyze boundary value problems of singularly perturbed differential difference equations with negative shift or delay. They used the invariant embedding technique and central and upwind finite difference discretization for the secondand first-order derivatives, respectively, and proved the stability and convergence of their method. An exponentially fitted difference scheme on a uniform mesh is accomplished by the method of integral identities to solve problems of the same type [33]. In this method, the authors used exponential basis functions and interpolating quadrature rules with weight and remainder terms in integral form. Various numerical methods have also been presented for solving singularly perturbed boundary value problems involving small shifts, including exponential methods based on piecewise analytical solutions of advection–reaction–diffusion operators [34], a fitted mesh B-spline collocation method [35], parameter–uniform numerical methods comprising a standard implicit finite difference scheme [36], ε-uniformly convergent non-standard finite differences [37] and ε-uniformly convergent fitted methods [38]. We devised a terminal boundary value technique for solving singularly perturbed delay differential equations with a boundary layer at the left end of the interval. By introducing a terminal point into the domain, the original problem is divided into inner and outer region problems. A terminal boundary condition in the implicit form is determined from the reduced problem, and the outer region problem with the implicit boundary condition is then solved as a two-point boundary-value problem. From the solution of the outer region problem, an explicit terminal boundary condition is obtained. The inner region problem is modified and solved as a twopoint boundary value problem using the obtained explicit terminal boundary condition. Finally, we combined the solutions of the inner region and outer region problems to obtain the approximate solution of the original problem. The method is iterative on the terminal point. We repeated the numerical scheme for various choices of the terminal point until the solution profiles did not differ materially from iteration to iteration. To validate the efficiency of the method, some model examples are solved. The stability and convergence of the scheme was also investigated.

2. Description of the method Consider a linear singularly perturbed two-point boundary value problem of the form: εy (x) + a(x)y (x − δ) + b(x)y(x) = f (x),

0≤x≤1 (1)

Subject to the interval and boundary conditions y(x) = φ(x),

−δ≤x≤0

y(1) = β,

(2) (3)

where ε is a small positive parameter (0 < ε  1), δ is delay parameter, a(x), b(x), f(x) and φ(x) are sufficiently smooth functions, and β is a known constant. For δ = 0, the problem (1)–(3) becomes a boundary value problem for singularly perturbed ordinary differential equations. The layer behaviour of the problem / 0 but under consideration is maintained only for δ = is sufficiently small (i.e. δ is of o(ε)). When the delay parameter δ exceeds the perturbation parameter, ε (i.e. δ is of O(ε)), the layer behaviour of the solution is no longer maintained; rather, the solution exhibits an oscillatory behaviour or is diminished. We considered the cases in which δ is of o(ε) (i.e.δ < ε). Now, we assume that a(x) ≥ M > 0 and ε − δa(x) > 0 throughout the interval [0,1], where M is some positive constant. Under these assumptions, (1) has a unique solution y(x), which in general displays a boundary layer in the neighbourhood of x = 0. Since δ is of o(ε) and the solution y(x) of the BVP (1)–(3) is sufficiently differentiable, by using Taylor’s series expansion, we obtain y (x − δ) ≈ y (x) − δy (x)

(4)

Substituting (4) into (1), we obtain an asymptotically equivalent two-point boundary value problem ε(1 − ξa(x))y (x) + a(x)y (x) + b(x)y(x) = f (x)

(5)

with y(0) = φ(0)

(6)

y(1) = β

(7)

where δ = ξε with ξ = O(1). The transition from Eq. (1) to Eq. (5) is admitted because of the condition that the delay parameter 0 < δ  1 is sufficiently small and is of o(ε). Let xp (0 < xp  1) be the terminal point or width or thickness of the boundary layer. It is well known from the singular perturbation theory (see [1,3]) that the

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reduced equation (i.e. Eq. (5) with ε = 0) is valid in the outer region. Hence, by putting ε = 0 in (5), we have the reduced equation a(x)y (x) + b(x)y(x) = r(x)

for

xp ≤ x ≤ 1



c1 y (xp ) + c2 y(xp ) = c3

to form a new differential equation for the inner region solution. By using (17), we transform Eq. (12) with y(x) = y(tε) = Y (t)

(8)

Now, evaluating (8) at x = xp and denoting c1 = a(xp ), c2 = b(xp ) and c3 = f(xp ) gives us (9)

y (tε) Y  (t) = ε ε

(19)

y (x) =

y (tε) Y  (t) = ε2 ε2

(20)

a(x) = a(tε) = A(t)

(21)

b(x) = b(tε) = B(t)

(22)

f (x) = f (tε) = F (t)

(23)

to the new inner region problem of the form:

(ε − δa(x))y (x) + a(x)y (x) + b(x)y(x) = f (x),

(1 − ξA(t))Y  (t) + A(t)Y  (t) + εB(t)Y (t) = εF (t),

with c1 y (xp ) + c2 y(xp ) = c3 and y(1) = β

(10)

(11)

and the inner region problem: (ε − δa(x))y (x) + a(x)y (x) + b(x)y(x) = f (x), 0 ≤ x ≤ xp

(12)

with y(0) = φ(0), and y(xp ) = γ (is obtained as described below) (13) First, solving the outer region problem (ε − δa(x))y (x) + a(x)y (x) + b(x)y(x) = f (x), xp ≤ x ≤ 1

(14)

with c1 y (xp ) + c2 y(xp ) = c3

(15)

and y(1) = β

(16)

we obtain solution y(x) over [xp ,1]. From the solution y(x) of the outer region problem (14)–(16) on the interval xp ≤ x ≤ 1, we obtain the value of y(xp ) that is the explicit terminal boundary condition and denote it by y(xp ) = γ. Now, in order to solve the inner region problem, consider inner region problem (12) and (13) with the assumption δ = ξε, δ = o(ε) and choose the transformation x t= (17) ε

(18)

y (x) =

We take (9) as the terminal boundary condition in implicit form. As the terminal point xp is common to both the inner and outer regions, then the inner and outer region problems are defined on 0 ≤ x ≤ xp and xp ≤ x ≤ 1, respectively. That is, the outer region problem: xp ≤ x ≤ 1

3

0 ≤ t ≤ tp

(24)

and the boundary condition for (24) is determined by (13) and (18). Y (0) = φ(0) = φ0

(25)

and Y (tp ) = y(xp ) = γ

(26)

where tp = xp /ε. We solve the new inner region problem (24)–(26) to obtain the solutions over the interval 0 ≤ t ≤ tp . To solve the two-point boundary value problems given in Eqs. (14)–(16) (outer region problem) and (24)–(26) (inner region problem), we make use of a second-order classical finite difference scheme. In fact, any standard analytical or numerical method can be used. Finally, we combine the solutions of both the inner region defined in 0 ≤ x ≤ xp and the outer region defined in xp ≤ x ≤ 1 problems to obtain the approximate solution of the original problem (1)–(3) over the interval 0 ≤ x ≤ 1. We repeat the process (numerical scheme) for various choices of xp (the terminal point), until the solution profiles do not differ materially from iteration to iteration. For the computational point of view, we use an absolute error criterion, namely    m+1  (x) − ym (x) ≤ σ, 0 ≤ x ≤ xp y where ym (x) = the solution for the mth iterate of xp and σ = the prescribed tolerance bound. To describe the scheme for the outer region problem (14)–(16), we divide [xp ,1] into N equal parts, each of length h, xp = x0 < x1 < x2 < · · · < xN = 1 and we have xi = xp + ih, i = 0, 1, 2, . . ., N. For convenience, let a(xi ) = ai , b(xi ) = bi , f(xi ) = fi , y(xp ) = y0 , y(xi ) = yi . Now, applying the classical central finite difference scheme to

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(14)–(16), we obtain the three-term recurrence relation

where

Ei yi−1 − Fi yi + Gi yi+1 = Hi ,

Ei =

i = 0, 1, . . ., N − 1 (27)

where (ε − δai ) ai − h2 2h 2(ε − δai ) Fi = − bi h2 (ε − δai ) ai Gi = + 2 h 2h H i = fi Ei =

(28) 3. Lower bound of the terminal point tp

2hc2 2hc3 y 0 + y1 − c1 c1

(29)

where c1 , c2 and c3 are defined in (9). Using (29) in the first equation of the recurrence relation (27) at i = 0, we obtain   2hc2 − F0 + E0 y0 + (E0 + G0 )y1 c1 = H0 +

(32)

The Thomas algorithm is used to solve the tridiagonal system (31).

Eq. (27) gives a system of N equations with N + 1 unknown y−1 to yN−1 . To eliminate the unknown y−1 , we use the implicit boundary condition (15); then, by employing the second-order central difference approximation in it, we obtain y−1 =

(1 − ξAi ) Ai − h2 2h − ξA (1 ) i Fi = − εBi h2 (1 − ξAi ) Ai Gi = + h2 2h  Hi = εFi

2hc3 E0 c1

To gain further insight into the choice of tp , the terminal point of the boundary layer region, consider the problem (1)–(3). Further, choose tp such that tp  1/ε. Now, using Eq. (5) and assuming that δ  ε and ε − δa(x) > 0, we get (ε − δa(x))y (x) + a(x)y (x) + b(x)y(x) = f (x)

(33)

Using the stretching transformation (17) and taking the limit as ε → 0, we obtain   δa(0) 1− (34) Y  (t) + a(0)Y  (t) = 0 ε where Y(t) = y(tε) and the boundary conditions are Y (0) = φ(0)

and

Y (tp ) = γ

(35)

Now, solving the two-point boundary value problem (34) and (35) analytically, we obtain the solution (30)

Now, Eqs. (27) and (30) give N by an N tri-diagonal system, which can be easily solved with the Thomas algorithm. Similarly, to set up the difference equation for the inner region problem (24)–(26), we divide the interval 0 ≤ t ≤ tp into N subintervals of equal mesh length h = (tp − 0)/N with mesh points 0 = t0 < t1 < t2 , . . ., < tN = tp . Again applying the second-order classical finite difference scheme to (24)–(26), we obtain the three-term recurrence relation

Y (t) = m1 + m2 emt

(36)

where m = (− εa(0))/(ε − δa(0)), m2 = (γ − φ(0))/ (e(−εa(0)tp )/(ε−δa(0)) − 1), and m1 = φ(0) − m2 . As suggested by Hsiao and Jordan [39] and Lorenz [40], tp can be determined by taking the inequality   −εa(0) exp (37) tp < ε ε − δa(0) Taking the natural logarithm (ln) of both sides of (37), we get −εa(0)tp < ln ε ε − δa(0)

(38)

Now, rearranging (38) gives Ei yi−1

− Fi yi

+ Gi yi+1

=

Hi ,

i = 1, 2, . . ., N − 1 (31)

tp ≥

δa(0) − ε ln ε εa(0)

(39)

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For ε = 10−μ , we can obtain a crude estimate of the lower bound of tp from  1 − δa(0)10μ   1 − δa(0)10μ 

tp ≥μ

a(0)

ln(10) ≈ 2.3μ

a(0)

.

(40)

4. Stability and convergence analysis Theorem 1. Under the assumptions (ε − δa(x)) > 0, a(x) ≥ M > 0 and b(x) < 0, ∀x ∈ [0, 1], the solution to the system of the difference equations (27), together with the given boundary conditions, is unique and satisfies yh,∞ ≤ 2M −1 f h,∞ + (|φ0 | + |β|)

(41)

where ·h,∞ is the discrete l∞ − norm, given by xh,∞ = max {|xi |}. 0≤i≤N

5

Since (ε − δM) > 0, ah,∞ ≥0, bi < 0 and |wi | ≥0 ∀i, i = 1, 2, . . ., N − 1, for inequality (43) to hold, we must have wi = 0 ∀i, i = 1, 2, . . ., N − 1. This implies the uniqueness of the solution of the tridiagonal system of difference equations (27). For linear equations, the existence is implied by uniqueness. Now, to establish the estimate, let wi = yi − li , where yi satisfies difference equations (27), the boundary conditions and li = (1 − ih)φ0 + (ih)β, then w0 = wN = 0 and wi , fies

i = 1, 2, . . ., N − 1 satis-

Proof. Let Lh (.) denote the difference operator on the left-hand side of Eq. (27) and wi be any mesh function satisfying

Lh (wi ) = fi

Lh (wi ) = f

|wn | = wh,∞ ≥ |wi | ,

By rearranging the difference scheme (27) and using non-negativity of the coefficients Ei , Fi and Gi , we obtain

Then, summing (42) from i = n to N − 1 and using the assumption on a(x), gives

Fi |wi | ≤ |Hi | + Ei |wi−1 | + Gi |wi+1 |

−

Now, on the assumption (ε − δa(x)) > 0 and a(x) ≥ M > 0, the definition of l∞ -norm and manipulating the above inequality, we obtain   |wi+1 | − 2 |wi | + |wi−1 | (ε − δM) h2   |wi+1 | − |wi−1 | + ah,∞ + bi |wi | + |fi | ≥0 2h (42)

+

To prove the uniqueness and existence, let {ui } , {vi } be two sets of solution of the difference equation (27) satisfying boundary conditions. Then, wi = ui − vi satisfies Lh (wi ) = fi where fi = 0 and w0 = wN = 0. Summing (42) over i = 1, 2, . . ., N − 1, we obtain −

(ε − δM) (ε − δM) |w1 | − |wN−1 | h2 h2   N−1  |w1 | − |wN−1 | − ah,∞ bi |wi | ≥0 + 2h i=1

Now, let i = 0, 1, . . ., N.

(ε − δM) (|wn | − |wn−1 | + |wN−1 |) h2 M M |wn | (|wn−1 | − |wN−1 |) − − 2h 2h N−1 

N−1 

i=n

i=n

bi |wi | +

|fi | ≥0

(44)

Inequality (44), together with the condition on b(x), implies that N−1

N

i=n

i=0

  M |wn | ≤ h |fi | ≤ h |fi | ≤ f h,∞ , 2 i.e., we have |wn | ≤ 2M −1 f h,∞

(45)

We also have yi = wi + li , yh,∞ = max {|yi |} 0≤i≤N

(43)

≤ wh,∞ + lh,∞

(46)

≤ |wn | + lh,∞ .

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6

Table 1 Numerical results for Example 5.1, ε = 10−3 , δ = 0.5ε. x

tp = 20 y(x)

tp = 15 y(x)

tp = 10 y(x)

tp = 5 y(x)

Exact solution

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0100 0.0150 0.0200 0.1000 0.2000 0.4000 0.6000 0.8000 0.9000 1.0000

1.0000000 0.3171426 0.3603879 0.3666117 0.3678324 0.3683816 0.3708603 0.3733556 0.3758674 0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

1.0000000 0.4528503 0.3792739 0.3697083 0.3687953 0.3690566 0.3714947 0.3739937

1.0000000 0.4533947 0.3798923 0.3703373 0.3694263 0.3696884 0.3721293

1.0000000 0.4538989 0.3804652 0.3709202 0.3700112 0.3702742

0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

1.0000000 0.4538692 0.3803509 0.3707303 0.3697488 0.3699358 0.3717605 0.3736230 0.3754949 0.4067525 0.4495085 0.5489761 0.6704540 0.8188125 0.9048827 1.0000000

Underline is used to indicate the numerical solutions obtained at the terminal points.

To complete the estimate, we have to find the bound on li lh,∞ = max {|li |} ≤ max {|(1 − ih)| |φ0 | 0≤i≤N

0≤i≤N

+ |ih| |β|} ≤ max {(1 − ih) |φ0 | + (ih) |β|} , 0≤i≤N

From Eqs. (45)–(47), we obtain the estimate yh,∞ ≤ 2M −1 f h,∞ + (|φ0 | + |β|) This theorem implies that the solution to the system of difference equations (27) is uniformly bounded, independently of mesh size h and the perturbation parameter ε. Thus, the scheme is stable for all step sizes.

i.e., we have lh,∞ ≤ |φ0 | + |β| .

(47)

Corollary 1. Under the conditions of Theorem 1, the error ei = y(xi ) − yi between the solution y(x) of the

Table 2 Numerical results for Example 5.1, ε = 10−4 , δ = 0.5ε. x

tp = 20 y(x)

tp = 15 y(x)

tp = 10 y(x)

tp = 5 y(x)

Exact solution

0.00000 0.00010 0.00020 0.00030 0.00040 0.00050 0.00100 0.00150 0.00200 0.10000 0.20000 0.40000 0.60000 0.80000 0.90000 1.00000

1.0000000 0.3181581 0.3612116 0.3670360 0.3678237 0.3679338 0.3682020 0.3684702 0.3687384 0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 1.0000000

1.0000000 0.4709184 0.3873736 0.3711528 0.3689013 0.3685967 0.3685541 0.3685541

1.0000000 0.4707524 0.3871895 0.3709686 0.3687171 0.3684126 0.3683699

1.0000000 0.4705484 0.3869631 0.3707419 0.3684904 0.3681859

0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 0.0000000

0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 0.0000000

0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 0.0000000

1.0000000 0.4534718 0.3795465 0.3695746 0.3682570 0.3681105 0.3682659 0.3684501 0.3686343 0.4065880 0.4493469 0.5488281 0.6703334 0.8187389 0.9048420 1.0000000

Underline is used to indicate the numerical solutions obtained at the terminal points.

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Table 3 Numerical results for Example 5.2, ε = 10−3 , δ = 0.5ε.

7

Table 5 Numerical results for Example 6.1, ε = 10−3 , δ = 0.5ε.

x

tp = 20 y(x)

tp = 15 y(x)

tp = 10 y(x)

tp = 5 y(x)

x

tp = 20 y(x)

tp = 15 y(x)

tp = 10 y(x)

tp = 5 y(x)

0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0100 0.0150 0.0200 0.1000 0.2000 0.4000 0.6000 0.8000 0.9000 1.0000

1.0000000 0.3716997 0.2869179 0.2756975 0.2744568 0.2745652 0.2762854 0.2780575 0.2797985 0.3038107 0.3383586 0.4267997 0.5516247 0.7323793 0.8531665 1.0000000

1.0000000 0.3719800 0.2872367 0.2760220 0.2747825 0.2748914 0.2766135 0.2783891

1.0000000 0.3723023 0.2876031 0.2763947 0.2751564 0.2752659 0.2769902

1.0000000 0.3725916 0.2879317 0.2767291 0.2754918 0.2756018

1.0000000 0.4568711 0.3833696 0.3734207 0.3720739 0.3718956 0.3721293

1.0000000 0.4554691 0.3817777 0.3718032 0.3704530 0.3702742

0.3038107 0.3383586 0.4267997 0.5516247 0.7323793 0.8531665 1.0000000

0.3038107 0.3383587 0.4267997 0.5516248 0.7323793 0.8531665 1.0000000

1.0000000 0.4596118 0.3864810 0.3765825 0.3752426 0.3750651 0.3753012 0.3755843 0.3758674 0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

1.0000000 0.4582375 0.3849208 0.3749970 0.3736535 0.3734757 0.3737106 0.3739937

0.3038107 0.3383586 0.4267997 0.5516247 0.7323793 0.8531665 1.0000000

0.00000 0.00100 0.00200 0.00300 0.00400 0.00500 0.01000 0.01500 0.02000 0.10000 0.20000 0.40000 0.60000 0.80000 0.90000 1.00000

0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

0.4071563 0.4499552 0.5495225 0.6711221 0.8196300 0.9057869 1.0000000

Underline is used to indicate the numerical solutions obtained at the terminal points.

Underline is used to indicate the numerical solutions obtained at the terminal points.

continues problem and the solution yi of the discretized problem, with boundary conditions, satisfies the estimate eh,∞ ≤ 2M −1 τh,∞ where

(48)





h2 a(x)  (3)  y (x) xi−1 ≤x≤xi+1 6  2   εh   + max (ε − δa(x)) y(4) (x) xi−1 ≤x≤xi+1 12

|τi | ≤

Proof. Truncation error τ i is given by

max

(49)

Table 4 Numerical results for Example 5.2, ε = 10−4 , δ = 0.5ε.

yi+1 − 2yi + yi−1 τi = (ε − δa(x)) h2   yi+1 − yi−1  + a(x) − yi 2h



− yi



Table 6 Numerical results for Example 6.1, ε = 10−4 , δ = 0.5ε.

x

tp = 20 y(x)

tp = 15 y(x)

tp = 10 y(x)

tp = 5 y(x)

x

tp = 20 y(x)

tp = 15 y(x)

tp = 10 y(x)

tp = 5 y(x)

0.00000 0.00010 0.00020 0.00030 0.00040 0.00050 0.00100 0.00150 0.00200 0.10000 0.20000 0.40000 0.60000 0.80000 0.90000 1.00000

1.0000000 0.3313265 0.2408234 0.2290035 0.2292476 0.2315017 0.2446982 0.2588526 0.2738332 0.3027937 0.3372552 0.4255062 0.5501304 0.7307238 0.8514816 1.0000000

1.0000000 0.3425767 0.2535924 0.2420007 0.2423802 0.2447798 0.2587352 0.2736963

1.0000000 0.3544611 0.2670812 0.2557299 0.2562526 0.2588054 0.2735594

1.0000000 0.3668457 0.2811385 0.2700383 0.2707104 0.2734225

1.0000000 0.4707524 0.3871895 0.3709686 0.3687171 0.3684126 0.3683699

1.0000000 0.4705484 0.3869631 0.3707419 0.3684904 0.3681859

0.3027937 0.3372553 0.4255062 0.5501304 0.7307238 0.8514816 1.0000000

0.3027937 0.3372553 0.4255062 0.5501304 0.7307238 0.8514816 1.0000000

1.0000000 0.4710844 0.3875577 0.3713371 0.3690856 0.3687811 0.3687384 0.3687384 0.3687384 0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 0.0000000

1.0000000 0.4709184 0.3873736 0.3711528 0.3689013 0.3685967 0.3685541 0.3685541

0.3027937 0.3372553 0.4255062 0.5501304 0.7307238 0.8514816 1.0000000

0.00000 0.00010 0.00020 0.00030 0.00040 0.00050 0.00100 0.00150 0.00200 0.10000 0.20000 0.40000 0.60000 0.80000 0.90000 1.00000

0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 1.0000000

0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 1.0000000

0.4066914 0.4494485 0.5489215 0.6704094 0.8187855 0.9048674 1.0000000

Underline is used to indicate the numerical solutions obtained at the terminal points.

Underline is used to indicate the numerical solutions obtained at the terminal points.

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Fig. 1. Inner layer solutions of Example 5.1 for ε = 0.001, δ = 0.1ε and different terminal points.

h2 a(x)  (3)  |τi | ≤ max y (x) xi−1 ≤x≤xi+1 6  2   εh  (4)  + max (ε − δa(x)) y (x) xi−1 ≤x≤xi+1 12

Estimate (48) establishes the convergence of the difference scheme for the fixed values of the parameter ε. (50) 5. Numerical examples

One can easily show that the error ei satisfies

To validate the efficiency of the method, we applied it to two linear examples and one nonlinear example.

Lh (e(xi )) = Lh (y(xi )) − Lh (yi ) = τi , i = 1, 2, . . ., N − 1

(51)

and e0 = eN = 0. Then, Theorem 1 implies that eh,∞ ≤ 2M −1 τh,∞ .

(52)

Consider the singularly perExample 5.1. turbed delay differential equation with left layer εy (x) + y (x − δ) − y(x) = 0 ; x ∈ [0, 1], with y(0) = 1 and y(1) = 1.

Fig. 2. Inner layer solutions of Example 5.1 for ε = 0.0001, δ = 0.1ε and different terminal points.

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Fig. 3. Inner layer solutions of Example 5.2 for ε = 0.001, δ = 0.1ε and different terminal points.

The exact solution is given by (1 − em2 )em1 x + (em1 − 1)em2 x em1 − em2 √ where m1 =√(−1 − 1 + 4(ε − δ))/(2(ε − δ)) and m2 = (−1 + 1 + 4(ε − δ))/(2(ε − δ)). Numerical results are presented in Tables 1 and 2 for ε = 10−3 and ε = 10−4 , respectively. y(x) =

εy (x) + e−0.5x y (x − δ) − y(x) = 0,

0≤x≤1

with boundary conditions y(0) = 1, − δ ≤ x ≤ 0, y(1) = 1. The exact solution of the problem is not known. Numerical results are presented in Tables 3 and 4 for ε = 10−3 and ε = 10−4 , respectively. 6. Nonlinear problems

Example 5.2. Consider the following singularly perturbed variable coefficient and non-homogenous delay differential equation:

To solve nonlinear singular perturbation problems, we used the method of quasi-linearization.

Fig. 4. Inner layer solutions of Example 5.2 for ε = 0.0001, δ = 0.1ε and different terminal points.

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Fig. 5. Inner layer solutions of Example 6.1 for ε =0.001, δ = 0.1ε and different terminal points.

Example 6.1. Consider the following non-linear singularly perturbed delay differential equation: εy (x) + y(x)y (x − δ) − y(x) = 0

The exact solution of the problem is not known. Numerical results are presented in Tables 5 and 6 for ε = 10−3 and ε = 10−4 , respectively. 7. Discussion and conclusions

under the interval and boundary conditions y(x) = 1,

− δ ≤ x ≤ 0,

y(1) = 1

The quasilinear form of this example is εy (x) + y (x − δ) − y(x) = 0; y(x) = 1, − δ ≤ x ≤ 0, y(1) = 1

A terminal boundary-value technique has been presented for solving singularly perturbed delay differential equations whose solutions exhibits boundary layer behaviour. The method is iterative on the terminal point xp and the process is to be repeated for different values of xp (the terminal point which is not unique), until the

Fig. 6. Inner layer solutions of Example 6.1 for ε = 0.0001, δ = 0.1ε and different terminal points.

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solution profile stabilizes in both the inner and outer regions. The present method has been implemented on two linear and one nonlinear problem with left-end boundary layer, by taking δ = 0.1ε, δ = 0.5ε and different values of ε. The numerical results have been tabulated and compared with the exact solutions. Although the solutions are computed at all the points with mesh size h only a few values have been reported. It can be observed from the tables (Tables 1–6) and figures (Figs. 1–6) that the present method approximates the exact solution very well. In fact, the method helps us not only to get good results but also to know the behaviour of the solution in the boundary layer/inner region with h ≥ ε where the existing numerical methods fail to give good results (See Figs. 1–6). The method is simple, easy and efficient technique for solving singularly perturbed delay differential equations.

References [1] R.E. O’Malley, Introduction to Singular Perturbations, Academic Press, New York, 1974. [2] R.E. O’Malley, Singular Perturbation Methods for Ordinary Differential Equations, Springer, New York, 1991. [3] A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973. [4] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981. [5] J.D. Cole, J. Kevorkian, Perturbation Methods in Applied Mathematics, Springer, New York, 1979. [6] C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978. [7] W. Eckhaus, Matched Asymptotic Expansions and Singular Perturbations, Elsevier North-Holland, Amsterdam, 1973. [8] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA, 1975. [9] R. Bellman, Perturbation Techniques in Mathematics, Physics and Engineering, Holt, Rinehart & Winston, New York, 1964. [10] E.P. Doolan, J.J.H. Miller, W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980. [11] H. Goering, A. Felgenhauer, G. Lube, H.G. Roos, L. Tobiska, Singularly Perturbed Differential Equations, Akademic Verlag, Berlin, 1983. [12] P.W. Hemker, A Numerical Study of Stiff Two Point Boundary Problems, MCT 80, Mathematical Centre, Amsterdam, 1977. [13] P.W. Hemker, J.J.H. Miller (Eds.), Numerical Analysis of Singular Perturbation Problems, Academic Press, New York, 1979. [14] J.J.H. Miller (Ed.), Application of Advanced Computational Methods for Boundary and Interior Layers, Boole Press, Dublin, 1993. [15] J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, River Edge, NJ, 1996. [16] D.Y. Tzou, Macro-to-microscale Heat Transfer, Taylor & Francis, Washington, DC, 1997.

11

[17] D.D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys. 61 (1989) 41–73. [18] D.D. Joseph, L. Preziosi, Addendum to the paper heat waves, Rev. Mod. Phys. 62 (1990) 375–391. [19] M.W. Derstine, H.M. Gibbs, F.A. Hopf, D.L. Kaplan, Bifurcation gap in a hybrid optical system, Phys. Rev. A 26 (1982) 3720–3722. [20] Q. Liu, X. Wang, D. De Kee, Mass transport through swelling membranes, Int. J. Eng. Sci. 43 (2005) 1464–1470. [21] M. Bestehorn, E.V. Grigorieva, Formation and propagation of localized states in extended systems, Ann. Phys. (Leipzig) 13 (2004) 423–431. [22] T.A. Burton, Fixed points, stability, and exact linearization, Nonlinear Anal. 61 (2005) 857–870. [23] X. Liao, Hopf and resonant codimension two bifurcations in van der Pol equation with two time delays, Chaos Soliton Fract. 23 (2005) 857–871. [24] M.C. Mackey, L. Glass, Oscillations and chaos in physiological control systems, Science 197 (1977) 287–289. [25] M. Wazewska-Czyzewska, A. Lasota, Mathematical models of the red cell system, Mat. Stosow. 6 (1976) 25–40. [26] C.G. Lange, R.M. Miura, Singular-perturbation analysis of boundary-value problems for differential–difference equations: (ii) rapid oscillations and resonances, SIAM J. Appl. Math. 45 (1985) 687–707. [27] C.G. Lange, R.M. Miura, Singular perturbation analysis of boundary-value problems for differential–difference equations: (v) small shifts with layer behavior, SIAM J. Appl. Math. 54 (1994) 249–272. [28] C.G. Lange, R.M. Miura, Singular perturbation analysis of boundary-value problems for differential–difference equations: (iii) turning point problems, SIAM J. Appl. Math. 45 (1985) 708–734. [29] C.G. Lange, R.M. Miura, Singular perturbation analysis of boundary-value problems for differential–difference equations: (vi) small shifts with rapid oscillations, SIAM J. Appl. Math. 54 (1994) 273–283. [30] C.G. Lange, R.M. Miura, Singular-perturbation analysis of boundary-value problems for differential–difference equations, SIAM J. Appl. Math. 42 (1982) 502–531. [31] M.K. Kadalbajoo, K.K. Sharma, Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior, Electron. Trans. Numerical Anal. 23 (2006) 180–201. [32] M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl. Math. Comput. 157 (2004) 11–28. [33] G.M. Amiraliyev, E. Cimen, Numerical method for a singularly perturbed convection–diffusion problem with delay, Appl. Math. Comput. 216 (2010) 2351–2359. [34] J.I. Ramos, Exponential methods for singularly perturbed ordinary differential–difference equations, Appl. Math. Comput. 182 (2006) 1528–1541. [35] M.K. Kadalbajoo, D. Kumar, Fitted mesh B-spline collocation method for singularly perturbed differential–difference equations with small delay, App. Math. Comput. 204 (2008) 90–98. [36] D. Kumar, M.K. Kadalbajoo, A parameter-uniform numerical method for time-dependent singularly perturbed differential–difference equations, Appl. Math. Model. 35 (2011) 2805–2819. [37] K.C. Patidar, K.K. Sharma, Uniformly convergent nonstandard finite difference methods for singularly perturbed

Please cite this article in press as: G. File, Y.N. Reddy. Terminal boundary-value technique for solving singularly perturbed delay differential equations, J. Taibah Univ. Sci. (2014), http://dx.doi.org/10.1016/j.jtusci.2014.01.006

+Model JTUSCI-57; No. of Pages 12 12

ARTICLE IN PRESS G. File, Y.N. Reddy / Journal of Taibah University for Science xxx (2014) xxx–xxx

differential–difference equations with delay and advance, Int. J. Numerical Meth. Eng. 66 (2006) 272–296. [38] M.K. Kadalbajoo, K.C. Patidar, K.K. Sharma, ε-Uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, Appl. Math. Comput. 182 (2006) 119–139.

[39] G.C. Hsiao, K.E. Jordan, Solutions to the Difference Equations of Singular Perturbation Problems, Academic Press, New York, 1979. [40] J. Lorenz, Combinations of Initial and Boundary Value Method for a Class of Singular Perturbation Problems, Academic Press, New York, 1979.

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